RATIONAL SERIES IN THE FREE GROUP AND THE CONNES OPERATOR
AARON LAUVE AND CHRISTOPHE REUTENAUER
Abstract. We characterize rational series over the free group by using an oper- ator introduced by A. Connes. We prove that rational Malcev–Neumann series posses rational expressions without simplifications. Finally, we develop an effec- tive algorithm for solving the word problem in the free skew field.
1. Introduction Fix a field k and a finite set of indeterminants X. The free commutative field k(X) is defined as the initial object in the category of (commutative) fields F with embeddings k[X]֒→F . It also has a well understood realization as the field of rational functions: k(X)= f/g | f, g ∈ k[X], g = 0 . The focus of this paper is the noncommutative counterpart to k(X), the free (skew) field k<(X>.) It has an analogous categorical definition involving embeddings of k X , but one can hardly say that k<(X>) is well understood. As an illustration, the reader may take a moment to verify that x − z−1 1 − yx −1 y − z + y−1 − z−1 1 − x−1y−1 −1 x−1 − z = 0 (1.1) without allowing the variables to commute. A commonly used realization of the free field, due to Lewin [Lew74], involves Malcev–Neumann series over the free group Γ(X) generated by X. These are noncommutative, multivariate analogs of Laurent series; we recall the details in Section 2. While the Lewin realization of the free field is powerful, it suffers from an incon- venient asymmetry. We illustrate with an identity of Euler [Car00]: xk = 0. Z k∈ −k k This looks absurd, but rewriting it as k≥0 x + x k≥0 x , we recognize the geometric series expansions of 1/(1 − x−1) and x/(1 − x), respectively. Adding these fractions together does indeed give zero. Of course, kJX ∪ X−1K is not a ring, so the mixing of series in x and x−1 is usually disallowed. Nevertheless, passing through steps such as this one is a useful technique for proving identities in the free field. (See Appendix A.)
Date: July 4, 2012. 1 2 AARONLAUVEANDCHRISTOPHEREUTENAUER
In this paper, we give Euler’s identity firm footing in the noncommutative setting. (See [BHS09, BR03] for more on the commutative setting.) Our main result in this vein is a characterization of which series over the free group Γ(X) have rational expressions (Theorem 3.1 of Section 3). The main ingredient in the proof is a Fredholm operator F on the free group that we describe in Section 2.3. This operator featured prominently in a conjecture of Alain Connes [Con94] that was proven in [DR97]. Our result extends the main result there. Briefly, an element a ∈ kΓ is rational if and only if its associated Connes operator [F, a] has finite rank. In this paper, we also tackle the problem of rendering a given rational expression into Malcev–Neumann form (Section 4). Our main result here, Theorem 4.9, is an effective algorithm for solving the word problem in k<(X>) . Recall that the word problem for free groups (when are two words equal in Γ(X)?) is decidable, even though this is famously not the case for all groups [Nov55, Boo58]. The analogous problem for the free field (when are two expressions equal in k<(X>) ?) was first considered in [Coh73] and taken up again in [CR99]. Here is a simple example: Are x−1 (1 − x)−1 and x−1 + (1 − x)−1 equal? The answer is, “yes,” as the reader may easily verify. While algorithms are presented in [Coh73, CR99], their complexity seems very high. We derive our algorithm from Fliess’ proof [Fli71] of the following fact: simplifications in rational expressions preserve rationality. Here, at last, we may say that the word problem is certainly not undecidable. Though the complexity of our algorithm could certainly be improved.
2. The Free Field, Rational Series, and the Connes Operator 2.1. The free field. The free field k<(X>) is defined as the initial object in the category of epic k X skew fields with specializations. (See [Coh95, Ch. 4], which also contains a realization in terms of full matrices over k X .) In what follows, we use Lewin’s realization in terms of formal series. 2.1.1. Operations on series over the free group. Let Γ = Γ(X) denote the free group generated by a finite set of noncommuting indeterminants X. Let kΓ denote the vector space with basis Γ and kΓ denote the formal series over Γ, i.e., functions a: Γ → k, which we may write as ω∈Γ(a, ω) ω or ω∈Γ aω ω as convenient. The support supp(a) of a series a is the set of its nonzero coefficients, {ω ∈ Γ | (a, ω) = 0}. The sum a + b of two series over Γ is well-defined and defined by (a + b, ω) = (a, ω) + (b, ω) for all ω ∈ Γ. The Cauchy product a b (or ab) of two series is well-defined if (ab, ω) := (a, α)(b, β) α,β∈Γ αβ =ω is a finite sum for all ω ∈ Γ. We will also have occasion to use the Hadamard product a ⊙ b, defined by (a ⊙ b, ω) := (a, ω)(b, ω) for all ω ∈ Γ. We define now a unary operation called the star operation: given a ∈ kΓ, we let a∗ denote the sum 1+ a + a2 + , if this is a well-defined element of kΓ; in other words, if for any ω ∈ Γ, there are only finitely many tuples (ω1,...,ωn) such that RATIONALSERIESANDTHECONNESOPERATOR 3
Γ ω = ω1 ωn and (a, ω1) (a, ωn) = 0. In this case, it is the inverse of 1 − a in k under the Cauchy product. 2.1.2. Rational series over the free group. Fix a total order < on Γ compatible with its group structure. (See Appendix C for an elementary example.) The Malcev– Neumann series (with respect to <) is the subset k((Γ)) ⊆ kΓ of series with well- ordered support. That is, a ∈ k((Γ)) if every nonempty set A ⊆ supp(a) has a minimum element. One shows that k((Γ)) is closed under addition, multiplication and taking inverses, i.e., k((Γ)) is a skew field. See [Coh95, Ch. 2.4], [Pas85, Ch. 13.2], or [Sak09, Ch. IV.4]. The inverse of a Malcev–Neumann series a is built in a
geometric series–type manner: if a = aoωo + ωo<ω aωω, then −1 ∗ −1 −1 −1 −1 −1 a = 1 − aω ωωo aoωo = ao ωo aω ωωo , (2.1) ωo<ω ωo<ω −1 −1 −1 with aω = −aωao and ωo being the minimum element of supp (a ). Lewin [Lew74] showed that the free field k<(X>) is isomorphic to the rational closure of k X in k((Γ)), i.e., the smallest subring of k((Γ)) containing k X and closed under addition, multiplication and taking inverses. An alternative proof, using Cohn’s realization of the free field in terms of full matrices, appears in [Reu99]. 2.2. Rational series over free monoids. The rational series in k((Γ)) introduced above are a particular case of rational series over an alphabet A. In this more general notion, one starts with polynomials on the free monoid kA∗ and builds expressions with (+, ×, (·)∗). (Here, for an expression p ∈ kA∗ without constant term, p∗ := 1+ p + p2 + .) Indeed, let X = {x | x ∈ X} represent formal inverses for the elements of X, and put A = X ∪ X. Then (2.1) allows us to replace the (·)−1 operation over Γ with the (·)∗ operation over A. We do so freely in what follows and exploit results from the latter theory that we collect here. ∗ 2.2.1. Hankel rank. Let S = w(S, w)w be a series over A . Given a nonempty word u ∈ A∗, we define the new series u ◦ S ∈ k A , a (right) translate of S, by ◦ u S := (S, w)wo. w : w=wou Letting ε denote the empty word, put ε ◦ S := (S, ε) and extend bi-linearly to view ◦ as a map ◦ : kA∗ ⊗ k A → k A . (See [Sak09, Ch. III.4] or [BR11, Ch. 1.5], where the image is denoted Su−1.) The Hankel rank of the series S, introduced in [Fli74], is the rank of the operator ◦ S : kA∗ → k A . We extend this construction to series over the free group Γ in Section 3.4. The following is classical. Theorem 2.1 (Fliess). A series S ∈ k A over a free monoid A∗ is rational if and only if its right translates {f ◦ S | f ∈ kA∗} span a finite dimensional subspace of k A . An easy corollary is that the Hadamard product preserves rationality. For a proof, see [BR11, Th. 1.5.5]. 4 AARONLAUVEANDCHRISTOPHEREUTENAUER
Theorem 2.2 (Sch¨utzenberger). If S, T are rational series, then S ⊙ T is rational. 2.2.2. Recognizable series. We recall some additional results on recognizable series that will be useful in Section 4. Fix a series S = w(S, w)w ∈ k A . Let L = supp(S). The language L is said to be recognizable if there is an au- tomaton1 that accepts L and no other words. For example, the machine in Figure 1 accepts supp(b + ac)∗; the input state is marked with an arrow and the output state is doubly circled. More generally, we may extend the notion of automata so that a b 0 1 c
Figure 1. An automaton recognizing the language supp(b + ac)∗. edge-labels carry coefficients other than 0 or 1, e.g., taking values in some semiring k, and speak about S being recognizable by a k-automaton. Theorem 2.3 (Sch¨utzenberger). A series S is rational iff it is recognizable. We say that S is representable, with order n, if there exist row and column vectors λ ∈ k1×n, ρ ∈ kn×1, and a monoid morphism : A∗ → kn×n satisfying (S, w)= λ (w)ρ for all w ∈ A∗. For example, a representation (λ, , ρ) of the series (b + ac)∗ is provided by 0 1 1 0 0 0 1 λ = 1 0 , (a)= , (b)= , (c)= , ρ = . 0 0 0 0 1 0 0 x Here, (x)ij = 1 iff there is an edge i −→ j in the automaton. The equivalence between recognizable and representable series is well-known. (In fact, the usage of “recognizable” in the literature for what we call here “representable” is common- place.) In [Fli74], Fliess shows that the least n for which S has a representation is the Hankel rank of S [BR11, Th. 2.1.6]. 2.3. The Connes operator. In his book Noncommutative Geometry, Connes [Con94] gives a new proof of a celebrated C∗-algebra result2 of Pimsner–Voiculescu [PV82] using the machinery of “Fredholm modules.” We recount the details that are relevant to the present work. 2.3.1. Reduced words. Recall that each element ω ∈ Γ has a unique reduced expres- −1 sion ω = x1 xn with xi ∈ X or xi ∈ X for all i and no pair (i, i+1) satisfying xixi+1 = 1Γ. We say that ω has length ℓ(ω)= n. ∗ We identify Γ with the subset of reduced words, red(X ∪ X) , within the free ∗ monoid: say ω ∈ (X ∪ X) is reduced if it contains no factor of the form xx or x x
1A finite state machine that processes words and transitions between states by reading letters one at a time [Sak09]. ∗ 2Namely, the reduced C -algebra of the free group does not contain nontrivial idempotents. RATIONALSERIESANDTHECONNESOPERATOR 5 for any x ∈ X. We often denote the inverse of an element ω ∈ Γ by ω, and put ω = ω, to reduce clutter in the expressions that follow. Finally, if ω ,...,ω are elements of . 1 r Γ, we write ω = ω1 ωn if the product is a reduced factorization of ω (in particular, each ωi is reduced). That is, ω = ω1 ωr in Γ and ℓ(ω)= ℓ(ω1)+ + ℓ(ωr). −1 Let ω = x1 xn (xi ∈ X or xi ∈ X) be the reduced expression for ω ∈ Γ\{1}. Its longest proper prefix ℘(ω) is the word x1 xn−1. Its (nonempty) suffixes comprise the set S(ω) = {xn, xn−1xn,...,x1 xn−1xn}. We take the codomains ∗ of ℘(·) and S(·)tobeΓor red(X ∪ X) , as convenient, with context dictating which is intended. 2.3.2. The Connes operator. Let G = (V, E) be the Cayley graph of Γ, the infinite tree with vertex set V = Γ and edge set E satisfying {υ,ω}∈E if and only if υ = ℘(ω) or ω = ℘(υ). (In this case, note that {αυ, αω} is again an element of E, for any α ∈ Γ.) On the linear space kG = kΓ ⊕ kE, define a bilinear operation Γ G Γ : k ⊗ kG → k as follows. For a = α∈Γ(a, α)α ∈ k and {υ,ω}∈E, put a ω := (a, α) αω and a {υ,ω} := (a, α) {αυ, αω}. α ∈Γ α ∈Γ Recall that a Fredholm operator between two normed vector spaces X and Y is a bounded linear map with finite dimensional kernel and cokernel. The Fredholm operator F: kG → kG that Pimsner–Voiculescu and Connes use is defined as follows. For all υ ∈V and {℘(ω),ω}∈E, 0 if υ = 1 F({℘(ω),ω}) := ω and F(υ) := {℘(υ), υ} otherwise. The reader may check that ker F = coker F = spank{1}. Definition 2.4. Given a ∈ kΓ, the Connes operator [F, a]: kG → kG is defined as the commutator F ◦ a − a ◦ F. We develop a formula for the action of [F, a] on an edge {℘(ω),ω}: [F, a] ℘(ω),ω = F ◦ a ◦ ℘(ω),ω − a ◦ F ◦ ℘(ω),ω = F aα α℘(ω), αω − a ω α∈Γ
= aα α℘(ω)+ aα αω − aααω + aααω α∈Γ α∈Γ α∈Γ α∈Γ ω∈ S(α) ω ∈ S(α) ω∈ S(α) ω∈ /S(α)
= aα α ℘(ω) − ω . (2.2) α∈Γ ω∈ S(α) Example 2.5. [F,xxy]{y,¯ y¯x¯} = xxy(¯y − y¯x¯)= xx − x, while [F,xyy]{y,¯ y¯x¯} = 0. After his proof of the Pimsner–Voiculescu result, Connes offers a conjecture about the rank of his operator [Con94, p. 342, Remark 3]. This was later proved by 6 AARONLAUVEANDCHRISTOPHEREUTENAUER
Duchamp and the second author [DR97]. A discrete topology analog of the conjec- ture goes as follows. Theorem 2.6. An element a of the Malcev–Neumann series k((Γ)) belongs to the rational closure of kΓ in k((Γ)) ⇐⇒ the operator [F, a] is of finite rank. That is, if the image [F, a]kG ⊆ kG is finite dimensional. This appears as Th´eor`eme 12 in [DR97], but the proof of the forward implication is not given. A complete proof is given here, as it follows from the proof of our first result below.
3. Rank of the Connes Operators By the rational closure of kΓ in kΓ, we mean the smallest subspace of kΓ that contains kΓ and is closed under the Cauchy product and the star operation. Here, we insist only that all products and stars involved are well-defined in the sense of Section 2.1.1. Our first result is the following generalization of Theorem 2.6. Theorem 3.1. A formal series a ∈ kΓ belongs to the rational closure of kΓ in kΓ ⇐⇒ the operator [F, a] is of finite rank. Sections 3.1 and 3.2 establish the necessary machinery for the proof. Sections 3.3 and 3.4 establish the forward and reverse implications, respectively. (See Appendix B for possible refinements and extensions.) As a first simplification, note that [F, ·]: kG → kG has finite rank if and only if F · Γ the restricted operator [ , ] kE : kE → k does. We focus on kE in what follows. 3.1. Elementary identities on Connes operators. Given a, b ∈ kΓ, it will be useful to express the operators [F, ab] and [F, a∗] in terms of [F, a] and [F, b].
Proposition 3.2. Suppose a, b ∈ kΓ are such that ab and a∗ are well-defined. Then [F, ab] and [F, a∗] are well-defined operators; moreover [F, ab] = [F, a]b + a[F, b], (3.1) [F, a∗] = a∗[F, a]a∗. (3.2) Proof of Identity (3.1). First, some seemingly inoccuous algebra: [F, ab]= Fab − abF = (Fab − aFb) + (aFb − abF) = [F, a]b + a[F, b]. Supposing a, b, ab ∈ kΓ, the operators [F, a], [F, b], [F, ab]: kE → kΓ are well-defined, in particular Fab and abF are well-defined operators by themselves. In order to conclude that the left- and right-hand sides of (3.1) are equal, we must check that aFb is also well-defined as an operator. This is handled in the next lemma. Lemma 3.3. If a, b, and ab belong to kΓ, then aFb is a well-defined operator from kE to kΓ. RATIONALSERIESANDTHECONNESOPERATOR 7
Γ Proof. Write a = α∈Γ aαα and b = β∈Γ bββ. Since ab ∈ k , we have ab = aα bβ γ, (3.3) γ∈Γ α,β∈Γ γ =αβ with γ=αβ aα bβ involving finitely many nonzero terms for each γ ∈ Γ. The effect of aFb on ℘(ω),ω ∈E is as follows: aFb ℘(ω),ω = aF bβ β℘(ω),βω β∈Γ
= a bβ β℘(ω)+ bβ βω β∈Γ β∈Γ ω∈ S(β) ω ∈ S(β)
= aαbβ αβ℘(ω)+ aαbβ αβω . (3.4) α,β∈Γ α,β∈Γ ω∈ S(β) ω ∈ S(β)
We claim that no υ ∈ Γ appears infinitely often in (3.4). Indeed, suppose {(αi, βi)} is an infinite sequence satisfying υ ∈ αi βi ℘(ω), αi βi ω for all i. Then there is an infinite subsequence {(αj, βj)} satisfying ∀j, υ = αj βj ℘(ω) or ∀j, υ = αj βj ω. The two cases are similar; we consider the latter. Let υ′ω denote the common value of ′ the elements of the subsequence, so αjβj = υ (∀j). Deduce from (3.3) that only a
finite number of the αj βj appear with nonzero coefficient aαj bβj . This proves the claim and the Lemma. Proof of Identity (3.2). If a ∈ kΓ is such that a∗ is well-defined, then a∗a, aa∗, a∗a∗ and a∗aa∗ are also well-defined, and (1 − a)a∗ = a∗(1 − a) = 1. Thus we may write [F, a∗] = Fa∗ − a∗F = a∗(1 − a)Fa∗ − a∗F(1 − a)a∗ = a∗Fa∗ − a∗aFa∗ − a∗Fa∗ + a∗Faa∗ = a∗[F, a]a∗. The above observations and Lemma 3.3 guarantee that the operators a∗Fa∗, a∗aFa∗ and a∗Faa∗ used in the intermediate steps are well-defined. Conclude that (3.2) holds whenever a∗ is well-defined. 3.2. Closed subspaces of kΓ. The following is a standard result on topological vector spaces. We include a proof for the sake of completeness. Lemma 3.4. Suppose V is a finite dimensional subspace of kΓ. Then V is closed with respect to the product topology on kΓ extending the discrete topology on k. Proof. In five easy steps. (i). Recall that in the discrete topology on k, a sequence converges if and only if it is eventually constant. Γ Γ (ii). Fix a ∈ k and write a = α(a, α) α. The product topology on k is such that a sequence {a ∈ kΓ} ⊆ V converges to a if and only if for each α ∈ Γ, there n exists Nα satisfying (an, α) = (a, α) for all n > Nα. 8 AARONLAUVEANDCHRISTOPHEREUTENAUER
(iii). Suppose that b1, . . . , bm is a basis for the finite dimensional subspace V ⊆ Γ k . Find elements α1,...,αm ∈ Γ satisfying
(b1, α1) (b2, α1) (bm, α1) (b1, α2) (b2, α2) (bm, α2) det . . . . = 0, . . .. . (b1, αm) (b2, αm) (bm, αm) which are guaranteed to exist by the independence of b1, . . . , bm. (n) (iv). Given {an}→ a as above, find constants si ∈ k satisfying (n) (n) (n) an = s1 b1 + s2 b2 + + sm bm (∀n). (3.5)
Put N = max {Nα1 ,...,Nαm }, with Nαi determined as in (ii). Let B be the matrix found in (iii). From (3.5), we have the system
(n) (b1, α1) (bm, α1) s1 (an, α1) . . . . . . .. . . = . . (b , α ) (b , α ) (n) (a , α ) 1 m m m sm n m Note that the right-hand side is constant for n > N. Specifically, it equals T [(a, α1) (a, αm)] . Since B is invertible, this system has a unique solution T [s1 sm] independent of n (for n > N). (v). We claim that
a = s1b1 + s2b2 + + smbm .
To see this, note that (3.5) reduces to an = s1b1 + + smbm for n > N, since the bi are linearly independent. It follows that an = a for all n > N and the lemma is proven.
3.3. Connes operators of rational elements. Given the preparatory results in Sections 3.1 and 3.2, we are ready to prove that Connes operators of rational ele- ments have finite rank.
Proof of Theorem 3.1 (Forward Implication). It is enough to induct on the (+, ×, ∗)-complexity of a rational series a ∈ kΓ. We check that if a ∈ kΓ then [F, a] has finite rank and that the finite rank condition is closed under +, × and ∗. (i): Closed under +. Suppose a = a′ + a′′ with [F, a′] and [F, a′′] finite rank. Then img [F, a] ⊆ img [F, a′]+ img [F, a′′], thus [F, a] has finite rank as well. (ii): Polynomials have finite rank. After (i) we may assume a is a monomial α ∈ Γ. Apply [F, a] to some e = ω eω ℘(ω),ω to get
eω α ℘(ω) − ω . ω : ω∈S(α) Conclude that the rank of [F, a] is precisely ℓ(α), the cardinality of S(α). RATIONALSERIESANDTHECONNESOPERATOR 9
(iii): Closed under ×. Given rational series a, b with [F, a], [F, b] finite rank, consider [F, ab]. By (3.1) we have img [F, ab] ⊆ img a[F, b]+ img [F, a]b. Since [F, b] has finite rank, we know that img a[F, b] is finite dimensional. To control img [F, a]b, we use the fact that img [F, a] is a closed subspace (Lemma 3.4). First, approximate b = β bβ β by the sequence
bn := bβ β . β : ℓ (β)≤n Given e ∈ kE, note that {bne} → be. This means [F, a]bne → [F, a]be and [F, a]be belongs to img [F, a] (since this space is closed). Conclude that img [F, a]b is finite dimensional. (iv): Closed under ∗. Suppose a ∈ kΓ with a∗ well-defined and [F, a] having finite rank. Use (3.2) to write [F, a∗] = a∗[F, a]a∗. The two arguments in (iii) may be combined to conclude that a∗[F, a]a∗ has finite rank. 3.4. Connes operators with finite rank. We first reprise Proposition 5 in [DR97] to bound the rank of [F, a] when a is rational, making the necessary changes to work over kΓ instead of k((Γ)). The completed proof of Theorem 3.1 then follows. Γ ◦ Γ Given ω ∈ Γ \ {1} and a = α∈Γ aα α ∈ k , we define the series ω a ∈ k by
◦ ω a := aααo (3.6) . α= αoω and extend this to define a bilinear map ◦ : k(Γ \ {1}) ⊗ kΓ → kΓ in the natural way. . (The symbol = has been defined in Section 2.3.1.) We now compare the Connes rank of [F, a] on kE to the Hankel rank of ◦ a on k(Γ \ {1}). Proposition 3.5. Fix |X| = n. If a ∈ kΓ belongs to the rational closure of kΓ in kΓ then the Connes and Hankel ranks of a are related by 1 rnk ◦ a ≤ rnk [F, a] ≤ 2n rnk ◦ a . 2n Remark. The upper bound on rnk [F, a] is tight. Consider X = {x,y} and the series a = x + y + x + y. Γ Proof. Fix ω = ℘(ω)x ∈ Γ and a = aα α ∈ k . The proof rests on the observation that [F, a]{℘(ω),ω} = (ω ◦ a) (x − 1). (3.7) (Compare (2.2).) To demonstrate the second inequality, we define auxillary linear Γ transformations Ta,y : kE → k for each y ∈ X ∪ X by
◦ {℘(ω),ω} → ω a (y − 1) = aα αo(y − 1). . α= αoω ◦ From (3.6) we have rnk Ta,y ≤ rnk a, and from (3.7) we have img [F, a] ⊆ ◦ y img Ta,y. So indeed, rnk [F, a] ≤ 2n rnk a. 10 AARONLAUVEANDCHRISTOPHEREUTENAUER
To demonstrate the first inequality, we build (partially defined) operators v Γ Ta,y∗ : kE → k for each y ∈ X ∪ X as follows:
∗ ∗ {℘(ω),ω} → [F, a]{℘(ω),ω} (−y )= − aα α ℘(ω) − ω y , α: ω∈S(α) ∗ 2 v with y = 1+ y + y + , as usual. The set on which any Ta,y∗ is defined is a v linear subspace of kE, and it acts linearly there. We claim that Ta,x∗ {℘(ω),ω} is well-defined. If so, we are done, because
v ∗ ∗ Ta,x∗ {℘(ω),ω} = − aα α ℘(ω) − ω x = aα αo (1 − x ) x α: ω∈S(α) . α= αoω
◦ = aα αo = ω a. . α= αoω v ◦ ◦ In other words, img a ⊆ y img Ta,y∗ , from which it follows that rnk a ≤ v 2n rnk T ∗ ≤ 2n rnk [F, a]. a,x Toward verifying the claim, we write [F, a] {℘(ω),ω} in standard series form, namely
[F, a] {℘(ω),ω} = (aαω − aα x ω) α x . S α: x ∈ (α) The expression we must analyze is v l Ta,x∗ {℘(ω),ω} = (aαω − aα x ω) α x x . l∈N, α∈Γ x ∈ S(α)
li Fix υ ∈ Γ and suppose {(αi, li) : i ∈ I} is a sequence satisfying αi xx = υ. For each l appearing in the sequence (li), there is a unique allowable αi, so we may order {(αi, li) : i ∈ I} by its second factor, saying (αi, li) < (αj, lj) whenever li < lj. Put l1 equal to the minimum l occuring. Since each αi satisfies x ∈ S(αi), the equalities li υ = αi xx are all reduced factorizations of υ. Finally, the chain l1 l2 α1 xx = α2 xx = (l1 < l2 < ) forces α2 to be a proper prefix of α1, and forces α3 to be a proper prefix of α2, and so on. Since the set of prefixes of υ is finite, I = I(υ) is a finite set and υ does not v occur infinitely often in Ta,x∗ . ∗ Proof of Theorem 3.1 (Reverse Implication). Identify Γ with red(X ∪ X) , then use Theorem 2.1 and Proposition 3.5.
4. Applications to k((Γ)) and k<(X>) It is easy to see that our proof of Theorem 3.1 works equally for Malcev–Neumann series, thereby giving a complete proof of Theorem 2.6. Here we discuss two further results stemming from the notion of rational series in kΓ. RATIONALSERIESANDTHECONNESOPERATOR 11
4.1. Expressions without simplifications, effectively. Recall that an element −1 ω = x1 xn (xi ∈ X or xi ∈ X) in Γ is called reduced if no pair (i, i + 1) satisfies xixi+1 = 1Γ. This is the starting point for our next notion: (∗-rational) expressions without simplification in kΓ. A sum of monomials in kΓ is an expression without simplification if each monomial involved is reduced. More generally, the sum a+b of two expressions without simplification is again an expression without simplification. (We are only concerned with multiplicative simplifications,so(xyx+x)+(y2 −x) is an expression without simplification.) Continuing, the product a b of two expressions without simplification is an expression without simplification if for every monomial α in the support of a and every monomial β in the support of b, the monomial αβ is reduced, i.e., ℓ(α)+ ℓ(β)= ℓ(αβ). If a is an expression without simplification and a∗ is well-defined, then a∗ is an expression without simplification if and only if a a is an expression without simplification. Example 4.1. As a simple example, note that xx∗ may be written x + x∗. For another example, consider (xyx)∗, which is a legal expression kΓ (and even in k((Γ)) n if 1 < y and xy < yx). On the one hand, it is equal to n≥0(xyx) , which has many simplifications; on the other hand, it is equal to xy∗x, which has none. 4.1.1. Rational series in kΓ. Given a well-defined ∗-rational series a ∈ kΓ, we know that the corresponding series of reduced words S ∈ k X ∪ X is rational [Ben69]. Our goal is an effective algorithm that yields a rational expression for S without simplification. It suffices to make a careful analysis of Fliess’ construction [Fli71, Th. 3]. (See [Sak09] for details on k-automata and Cartesian products.) Theorem 4.2. Given a well-defined ∗-rational expression, representing the series a ∈ kΓ, there exists an equivalent ∗-rational expression without simplification that is effectively computable. Proof. Given such a series a, we: first, produce a (k-)automaton that recognizes a; next, modify the automaton so that all of its paths are reduced words; and finally, construct a rational expression equivalent to a from the augmented automaton. (i): Fliess’ construction (simplified case). Let a be a well-defined ∗-rational series in kΓ, recognized by an automaton A over X ∪ X, with n nodes. Suppose that A y contains the transitions p −→x q −→x r for some x ∈ X ∪ X. For each transition o −→ p y in A (with y ∈ X ∪ X), add the transition o −→ r. (Note that we allow o to be the incoming edge to the automaton, with label 1, in which case we add another incoming edge o −→ r.) Repeat for all such triples p −→x q −→x r originally occurring in A. This builds an augmented automaton A′. Then repeat for any triple p −→x q −→x r in A′ not already appearing in A to build an automaton A′′. Continue in this way until all new reductions have been bypassed. We claim that this process terminates in a finite (not necessarily deterministic) automaton A˜. Indeed, at each point, one constructs a subgraph of the complete, labeled, directed graph on n nodes and 2|X| labels. This is a finite graph. (ii): Fliess’ construction (generic case). We modify the above construction to accommodate well-defined ∗-rational series a ∈ kΓ with more general coefficients. 12 AARONLAUVEANDCHRISTOPHEREUTENAUER
First, build a trim k-automaton A over X ∪ X, with n nodes, following its presen- tation as a rational series. αy βx γx Each triple o −→ p −→ q −→ r (x,y ∈ X ∪ X; α, β, γ ∈ k) will be handled in roughly the same way as above; more precisely, augment A to an automaton A′ by (αβγ)y adding the edge o −−−−→ r. The concern is that the procedure A′, A′′,... never terminates, or rather, results in an infinite automaton A(∞). This cannot happen, as we now argue. Evidently, at each step in the procedure, A(l) → A(l+1), no new nodes are added, and any added edge comes from choosing a letter from X ∪ X and a coefficient from k. Conclude that if A(∞) is an infinite graph, then there are nodes o, r in A(∞) and a letter x ∈ X ∪ X so that o −→ r has an infinite set of labels {ξ1x,ξ2x,... }, with ξi ∈ k. Now, since A was trim, the rational expression for a contains a coefficient computation (a, ω) that is an infinite sum, contradicting the assumption that the expression was well-defined. So we may assume that A(∞) is in fact a finite k-automaton A˜, constructible in finitely many steps. (iii): A rational expression without simplifications. It is well-known that the ∗ 3 ∗ reduced words red(X ∪ X) form a rational set, and thus χ := w∈red(X∪X) w is recognizable by an automaton X . In particular, each path accepted by X has a reduced word as its label. Form the Cartesian product A×X˜ . This automaton ∗ recognizes the Hadamard product S ⊙ χ = w∈red(X∪X) (S, w)w. That is, S itself. ˜ Moreover, each path in A×X has the property that its label is a reduced word. It follows that S, and hence a, has a rational expression without simplification. (For this last point, simply reverse the McNaughton–Yamada algorithm, which builds an automata from a given rational expression.4) 4.1.2. Rational series in k((Γ)). We may also deduce a Malcev–Neumann version of the above result. Lemma 4.3. Given any rational Malcev–Neumann series a over Γ, there exists a ∗-rational subset L of Γ satisfying supp(a) ⊆ L and L is well-ordered. Proof. The following facts are well-known [Pas85, Lem. 13.2.9], [Coh95, Ch. 2.4].
(+) If L1 and L2 are well-ordered, then L1 ∪ L2 is well-ordered. (×) If L1 and L2 are well-ordered, then L1L2 is also. ( ∗ ) If each ω ∈ L satisfies ω = 1 and L is well-ordered, then L∗ is also. Now construct L recursively, following the rational presentation of a. Corollary 4.4. Every ∗-rational Malcev–Neumann series over the free group has an equivalent ∗-rational expression without simplification that is effectively computable. Proof. Given a ∗-rational expression for a ∈ k((Γ)), let L be as in Lemma 4.3 and ∗ let χL be the corresponding rational series over red(X ∪ X) , well-ordered after its
3Combine, e.g., Proposition 6.2 and Theorem 6.1 in [Sak09, Ch. II]. 4See [MY60] or the proof of Theorem 5.1 in [Eil74, Ch. VII]. RATIONALSERIESANDTHECONNESOPERATOR 13 identification with Γ. Observe that in a trim automaton X for χL, each closed path has a label w > 1. Indeed, otherwise L contains supp(uw∗v) for some u, v ∈ ∗ (X ∪ X) , making L not well-ordered. Now follow Parts (ii) and (iii) in the proof of Theorem 4.2—replacing X by XL. Remark. Note that in the result above we assume that an expression for a ∈ k((Γ)) has been given with inverses computed according to (2.1). In particular, we needn’t determine the minimal element in the support of any subexpression within a. (How- ever, such decisions may also be made effectively, as we show in Proposition 4.8.) Example 4.5. Suppose x y y y 2 1 2 1 ∗ x x y (xyx) x x x xy∗. (4.1) x x 0 3 0 3 Here, the dashed lines indicate those edges added during the Fliess construction. Explicit computation of the Hadamard product with XL is suppressed. 4.2. The word problem in the free field. By the word problem in the free field, we mean the following: Given an expression for some a ∈ k<(X>) , determine if a = 0. This problem was first solved in [Coh73], using Cohn’s theory of full matrices. This solution was revisited in [CR99], where it was reformulated as an ideal-membership problem in a commutative ring (a solution is then possible using Gr¨obner bases and Buchberger’s algorithm). Here, we describe an alternative solution to the word problem, debarking from the Malcev–Neumann realization of k<(X>) . 4.2.1. Description of the algorithm. We need the following standard result from the theory of rational languages. Proposition 4.6. Given a rational series S ∈ k A of (Hankel) rank n, each word of length at least n in the support has a subword which is also in the support. Proof. Let w ∈ supp(S) be of length at least n, and let p0,...,pn be any prefixes of w of strictly increasing length. Given a representation of the series of dimension n, (λ, , ρ) say, consider the vectors λ (p0), . . . , λ (pn). Note that for some i, λ (pi) is a linear combination of the vectors with smaller index. Let si be the suffix of w corresponding to pi and multiply the linear combination on the right by (si). Deduce that λ (w) is a linear combination of the vectors λ (pjsi) for j < i. Finally, λ (w)ρ is nonzero (since w ∈ supp(S)), and hence so is one of the numbers λ (pjsi)ρ. Conclude that the subword pjsi of w is also in the support of S. Let S be a rational series over the doubled alphabet X ∪ X, with support included in the set of reduced words, and with well-ordered support (upon identifying the 14 AARONLAUVEANDCHRISTOPHEREUTENAUER free group with the set of reduced words). We would like a method of determining min supp(S). Our algorithm rests on the following key lemma. Lemma 4.7. There exists a bound N, depending on the rank n of the series S and on the cardinality of the alphabet, such that for each word w in supp(S) of length at least N, w has a factor of the form u = u1 un with each ui > 1 in the group ordering. Proposition 4.8. Given S as above, min supp(S) is effectively computable. Proof. We claim that for each word w in the support of S of length at least N (from Lemma 4.7), there exists a subword that is smaller for the group ordering and still in the support. This makes the problem finite: let N denote the set of words w of length at most N, together with all of their subwords; restrict to the set N∩supp(S), and compare these elements pairwise to find the minimum. (See Appendix C for one way to make these comparisons.) . To see the claim, suppose w = w0u1 unwn, with the ui > 1 as in Lemma 4.7 and the wi possibly empty. Note that deleting any subset I of the ui from w results in a subword of w that is smaller for the group ordering. (Indeed, a < b =⇒ uav < ubv for all a, b, u, v ∈ Γ.) Finally, put pj := w0u1 uj, and consider the subwords pjsi of w from the proof of Proposition 4.6. One of these will belong to supp(S). Thus determining whether supp(S)= ∅ is a finite problem, as we now indicate. Theorem 4.9 (The Word Problem). Given a ∈ k<(X>) , the problem of determining whether a = 0 has an effective solution. Proof. Given a rational expression for a, construct an equivalent ∗-rational expres- ∗ sion S over S ∈ red(X ∪ X) according to (2.1). This may be done effectively, after Proposition 4.8. Next, replace this S by a rational series without simplification, as in Corollary 4.4. Now the word problem is reduced to the well-known problem of deter- mining if a rational series in k X ∪ X is zero; this is solved using Sch¨utzenberger’s reduction algorithm (see [BR11, Sec. 2.3]). Keep at hand the representation (λ, , ρ) for S developed there. Finally, pick each word w of length at most N (from Lemma 4.7), and determine if any of its subwords belong to supp(S). (Recall (S, u) is simply computed as λ (u)ρ.) This completes the proof. 4.2.2. Proof of key lemma. After some preparatory results of Jacob [Jac80, Jac78], we will be ready to prove Lemma 4.7. Proposition 4.10 (Jacob, [BR11, Th. 3.5.1]). Given a rational series S ∈ k A , there exists an integer N1 such that for any word w ∈ supp(S), and for any factor- ization w = w0uw1 satisfying |u| ≥ N1, there exists a factorization u = pvs such ∗ that supp(S) ∩ supp(w0pv sw1) is infinite. Recall that an n × n matrix over k is called pseudo-regular if it belongs to a subgroup of the multiplicative semigroup of matrices. Equivalently, if it is similar g 0 to a matrix of the form with g ∈ GL ′ (k) and 0 ≤ n′ ≤ n. 0 0 n RATIONALSERIESANDTHECONNESOPERATOR 15 Proposition 4.11 (Jacob, [BR11, Exer. 3.5.1]). Let (λ, , ρ) be a representation of a rational series S ∈ k A . For all integers p, there is a bound N2, depending on p, the rank n of S, and the cardinality of A, so that if w is any word in supp(S) of length at least N2, then w has consecutive factors u1, . . . , up with all (ui) pseudo-regular (and having a common kernel and image). Proof of Lemma 4.7. Let S be a rational series whose support is included in ∗ red(X ∪ X) . We may assume that supp(S) is well-ordered, upon identifying Γ(X) ∗ with red(X ∪ X) . Fix some representation (λ, , ρ) of S of dimension n. Note that if u, v′, v′′ are elements of the free group with u< 1, then supp(v′u∗v′′) has no smallest element (since this is true for u∗). The same is true even for each infinite subset of supp(v′u∗v′′). From this we deduce that if u, v′, v′′ are words ∗ in red(X ∪ X) , then supp(S) ∩ supp(v′u∗v′′) cannot be infinite. Now, it follows from the proof in [BR11] of Proposition 4.10 that if (u) is pseudo-regular and v′uv′′ ∈ supp(S), then supp(v′u∗v′′) ∩ supp(S) is infinite. Thus, if (u) is pseudo- regular, and v′uv′′ ∈ supp(S), then u> 1. Using Proposition 4.11, we see that if a word w in supp(S) is long enough, then it has a factor u = u1 un such that each (ui) are pseudo-regular. Hence each ui > 1 and the lemma is proven. Remark. Jacob’s bounds are extremely large. In [Reu80], see also [Okn98, Th. n n 1.12], a common bound, better than N1 and N2, is given: N(n) = i=1 i + 1 . However, it is still rather large (and proveably too large for N1). It may be possible that one could further improve N2 as well. Appendix A. Using Euler’s Identity In the introduction, we have claimed that Euler’s identity is sometimes useful for verifying identities in the free skew field. Here we illustrate with (1.1) after first indicating how this identity was found. A.1. Skew-symmetry of quasi-Pl¨ucker coordinates. Given an n × n matrix A, a row index i, and a column index j, the (i, j)-quasideterminant is defined if the (n−1) × (n−1) submatrix Ai,j is invertible. In that case, we put i,j −1 |A|ij = Ai,j − Ai,[n]\j A A[n]\i,j . Here, subscripts represent row and column indices of A to keep (when building submatrices) and superscripts represent indices of A to delete. For example, a b c −1 b c a |A| = u v w = u − v w . 2,1 y z x x y z Gelfand and Retakh [GR91] introduced quasideterminants as a replacement for the determinant in noncommutative settings. Since that time, they have proven useful in a number of different settings [GKL+95, MR04, DFK11]. Specific to the present discussion is the use of quasi-Pl¨ucker coordinates to describe coordinate rings for quantum flags and Grassmannians [Lau10]. 16 AARONLAUVEANDCHRISTOPHEREUTENAUER Given an n × m matrix A with n Appendix B. Extensions & Open Problems We collect some possible extensions of the results presented in Section 3, as well as some open problems. B.1. Extending the main theorem. In what generality does Theorem 3.1 hold? Let us replace the field k by any topological ring R and say that the Cauchy product a b is well-defined for a, b ∈ RΓ if (ab, ω) := (a, α)(b, β) α,β∈Γ αβ =ω converges (in R) for all ω ∈ Γ. Problem 1. Do Proposition 3.2 and Lemma 3.3, which use the discrete topology on k, hold in this more general setting? RATIONALSERIESANDTHECONNESOPERATOR 17 Let R be a (not necessarily commutative) ring. A free R-module M has invariant basis number (IBN) if any two bases of M have the same cardinality. If every free R-module has this property, we say that R is an IBN ring.5 Problem 2. Does Lemma 3.4 hold if the field k is replaced by an IBN ring R? It would seem, then, that the proofs of Theorem 3.1 and Proposition 3.5 extend to the setting of topological IBN rings. Problem 3. Let R be any topological IBN ring. Is it true that an element a ∈ RΓ belongs to the rational closure of RΓ in RΓ iff the operator [F, a] is of finite rank? B.2. Rank and image of the Connes operators. Hand calculations suggest that the bounds given in Proposition 3.5 are far from tight beyond rank one. Problem 4. Find a family of examples showing the bound is tight beyond rank one, or find a different tight bound beyond rank one. Using (3.1) and (3.2), hand calculations further suggest that the image of [F, a] is spanned by [F, a]{℘(ω),ω} : ω ∈ S(a) , where S(a) denotes the (finite) set of all suffixes of all monomials used in a ∗-rational expression for a. Problem 5. Determine if this is indeed the case. Remark. A similar problem could be posed for the Hankel operator ◦ a, with the answer perhaps being found “between the lines” of existing proofs of the equivalence of rational and representable series. Appendix C. Lyndon Words in the Magnus Ordering The results in this appendix are not needed for any other result in the paper. Still, they were found during the search for a proof of Proposition 4.8, so this paper seems like the best place to share them. Our main result is Theorem C.2; the corresponding algorithm appears in Section C.4. C.1. The Magnus ordering. The standard proof that a free group may be ordered computes its lower central series, orders the corresponding (free abelian) quotient groups, then patches these orderings together to build an ordering of the free group [Sak09, Th. 7.3]. However, there is a very elementary means of ordering Γ(X) that requires only that the mapping M : Γ(X) → Z X given by x → 1+ x (for all x ∈ X) is an embedding. This is proven in [MKS66, Th. 5.6]. 5Commutative rings, noetherian rings and division rings are among the chief examples. See [Coh95, Ch. 1.4] for more information. 18 AARONLAUVEANDCHRISTOPHEREUTENAUER Let < be a total ordering on the free monoid X∗. (We use here the military ordering: length plus lexicographic.) Extend < to a total ordering of Z X by using the fact that Z is well-ordered. Putting x M(ω) < M(υ). Thus, e.g., x>Γ y because M(x)=1+1x + 0y and M(y)=1+0x + 1y. −1 −1 Similarly, the reader may verify that xy <Γ y x. Remark. While the ordering of Γ(X) just described is customarily called the Magnus ordering, the first proof we have been able to find belongs to Bergman [Ber90]. C.2. The subword function. Given an alphabet A, and words w, v ∈ A∗, the w subword function v returns the number of occurrences of v as a subword (i.e., xn n subsequence) of w [Eil74]. Note that l = , so this is a natural generalization x l of binomial coefficients to words. We make a connection to the Magnus transformation to extend the first argument of the subword function to the free group. If ω ∈ X∗, it is clear that ω M(ω)= v . v v ⊆ω ∗ ω See [Lot97, Prop. 6.3.6]. For ω ∈ Γ(X) and v ∈ X , define v to be the coefficient of v in M(ω). The extension of · from X∗ to Γ is continuous with respect to the v profinite topology on Γ.6 The following evident result will be useful in what follows. ∗ ′ Lemma C.1. Let 1 = v0 < v1 < be the total ordering of X . If ω,ω ∈ Γ(X), ′ N then ω <Γ ω if and only if there is some p ∈ with ω ω′ ω ω′ = for all i m1 m2 mr v = ℓ1 ℓ2 ℓr (ℓ1 >ℓ2 > >ℓr, mj ≥ 1), where the ℓi are Lyndon [Lot97, Ch. 5]. Our main result is a restriction of the search space in Lemma C.1 from all words in X∗ to Lyndon words. 6This is the coarsest topology such that all homomorphisms φ: Γ → G to finite, discrete groups are continuous. See [Hal50] or [MR93] for details. RATIONALSERIESANDTHECONNESOPERATOR 19 ∗ Theorem C.2. Let ℓ1 < ℓ2 < denote the Lyndon words in X , ordered with respect to the military ordering. If ω,ω′ ∈ Γ(X), then with respect to the Magnus ′ N ordering, ω <Γ ω if and only if there is some p ∈ with ω ω′ ω ω′ = for all i m1 mr ∗ Lemma C.3. If ℓ1 ℓr is the Lyndon decomposition of a word v ∈ X , then 1 ⊔⊔ m1 ⊔⊔ mr ℓ1 ⊔⊔ ⊔⊔ ℓr = v + (smaller words w.r.t. <). m1! mr! Proof of Theorem C.2. After Lemma C.1, we may compare ω and ω′ by comparing ω ω′ the functions · and · . Given v ∈ X∗, compute its Lyndon decomposition ℓ ℓ (with ℓ ≥ ℓ for all j < k). Using (C.1) and Lemma C.3, we have i1 ir ij ik · · · = − terms with u < v . v ℓ u j ij ∗ 7The recapitulation in [Lot97, Ch. 6] takes ω ∈ X , but the original theorem is stated in the generality that we need here. 20 AARONLAUVEANDCHRISTOPHEREUTENAUER Now, either M(ω) and M(ω′) agree on these words u, in which case they have no ω ω′ bearing on the comparison of v and v , or they differ, in which case the deter- ′ mination of whether or not ω <Γ ω has already been made. In any case, induction completes the proof. C.4. Effective computation. Finally, we indicate via example how to effectively ω ∗ compute ℓ for ω ∈ Γ(X) and ℓ a Lyndon word in X . Consider ℓ = xy. Note that w xy ⊔⊔ X∗ = w . ∗ xy w ∈X An automaton recognizing the series xy ⊔⊔ X∗ is shown in Figure 2. An equivalent x y 0 1 2 x,y x,y x,y Figure 2. 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(Aaron Lauve) Department of Mathematics & Statistics, Loyola University Chicago, 1032 W Sheridan Road, Chicago, IL 60660, USA E-mail address: [email protected] URL: http://www.math.luc.edu/∼lauve (Christophe Reutenauer) LaCIM, UniversiteduQu´ ebec´ a` Montreal,´ Case Postale 8888, succursale Centre-ville, Montreal´ (Quebec)´ H3C 3P8, CANADA E-mail address: [email protected] URL: http://www.lacim.uqam.ca/~christo/